A few interesting articles went by recently about tax rates. This New York Times blog post points out an ABC News article in which it appears that the reporter didn't understand the difference between marginal and average tax rates. This doesn't require higher math, but I found it tricky enough that when I suddenly understood the difference, it was definitely an a-ha moment. The articles explain the issue, but perhaps a visual aid would help more people reach the a-ha.

The red line is how much income you take home if there is no income tax. Unsurprisingly, pre-tax and after-tax income are the same if there is no tax. You make $20,000, you take home $20,000.

The yellow line is what you take home if you pay an average tax. I'm using the Wikipedia terminology here. In this example, if you earn $7825 or less, you pay 10%. If you earn between $7825 but less than $31,850, you pay %15. I.e., you are in the 15% bracket. More than 31,850, you pay 25%. I plotted after-tax pay at $1000 increments, and you can see clearly that when you go from $31,000 to $32,000, the amount of money you take home actually drops, because you went into a higher tax bracket. If you were making $31,800 and you accepted a $100 raise, your take-home pay would drop by thousands of dollars. It wouldn't make sense to take a raise unless it was many thousands of dollars, so that you would make up for the higher tax rate. This is how many people believe the US income tax system works. This is wrong.

The blue line shows a marginal tax. You pay 10% on the first $7825. Then you pay 15% on the amount between $7825 and $31,850. Then you pay 25% on the amount above $31,850. When you get a raise from $31,000 to $32,000, taking you into the next marginal tax bracket, your take-home pay increases. The only portion of your salary you are paying 25% tax on is the amount over $31,850, i.e., the last $150 dollars. This is the actual federal income tax system in the United States.

The tax brackets don't stop at 25% and $31,850. The chart below shows all of the brackets in the 2007 tax system, including the highest bracket, 35%, which starts at $349,700.

As you can see from the smooth curve of the blue line, it is always in the taxpayer's interest to earn more money. Each additional dollar of income always leads to increased after-tax income. Some people argue that the higher tax rates at the top reduce the incentive to earn more, because that 349,701st dollar is only worth 65 cents instead of 68 or 70 or 75, but I don't want to address that argument here. The point I'm trying to make is that the US has a marginal tax rate system, and earning an additional dollar never reduces after-tax income.

For analysis that moves beyond basic innumeracy, see this article at FiveThirtyEight and the stories it links to. The point there is that focusing on the rates and ignoring the bracket levels is also a mistake, and that maybe we should re-introduce higher rates for much higher income levels.